The United Nations publishes updated estimates and projections of the populations of all the world's countries, broken down by age and sex. These are widely used by international organizations, governments, the private sector and researchers, for example for climate modeling and for assessing progress towards the Millenium Development Goals. The UN's current projections are deterministic, but assessing uncertainty about population estimates and projections is important for policy-making and other purposes. We propose to develop a fully probabilistic population projection methodology. We will develop methods for probabilistic projection of fertility and mortality, taking account of within-country and between-country correlations. We will develop methods for probabilistic projection of international migration. We will develop methods for probabilistic population projections in countries with generalized sexually transmitted infectious disease epidemics, which require special methods because the demographic impact of such diseases is massive and different from most other diseases, being concentrated among the least vulnerable parts of the population, namely young sexually active adults. We will develop methods for reconstructing past populations with uncertainty from fragmentary data. We will produce publicly available software for implementing the new methods.